Snow fall on grass: Cause emitted particle to stick to hair particle system, ideally unconverted
Unfortunately, the short answer is "no, hair systems only interact with colliders, not other particle systems."

The slightly longer answer is that you might be able to make it LOOK like the particles are interacting with the hair if you set up a collision object that's within the hair. The particles could land on this invisible collision surface. Model the surface of your collision object to have some high-frequency bumps and you might be able to convince your viewer that the particles are landing in the hair.